Homework 11

(to be turned in on May 6 as part of WPR III Grade)

Sequences of Functions and Series

1. Consider continuous functions from $[0,1]$ to $\mathbb{R}$ satisfying $0\leq f_1\leq f_2\leq f_3 \leq \cdots$ which converge pointwise to f. Must f be continuous? Give a proof or a counterexample.

2. Prove convergence and find the limit, or prove that the series does not converge:

(1)

\begin{eqnarray} \text{(a)} &\quad 2+4+8+16+\cdots\\ \text{(b)} &\quad \frac1{10}+\frac1{100}+\frac1{1000}+\cdots\\ \text{(c)} &\quad \sum_{n=1}^\infty n^{\frac1n}\\ \text{(d)} &\quad \sum_{n=1}^\infty \frac{1}{n^3+\pi}\\ \text{(e)} &\quad \sum_{n=2}^\infty \frac{(-1)^n}{\ln n}. \end{eqnarray}

3. What are the possible values of rearrangements of the following series?

(2)

\begin{eqnarray} \text{(a)} &\quad \sum_{n=1}^\infty \frac{1}{2^n}\\ \text{(b)} &\quad \sum_{n=1}^\infty \frac{(-1)^n}{2^n}\\ \text{(c)} &\quad \sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}\\ \text{(d)} &\quad \sum_{n=1}^\infty \frac{1}{\sqrt{n}} \end{eqnarray}

The Lipschitz Condition

A function is said to be Lipschitz with Lipschitz constant C if $|f(x)-f(y)|\leq C|x-y|$ for all x,y in the domain. (See Wikipedia article on Rudolf_Lipschitz.)

4. What does the Lipschitz condition say about the secant lines of f?

5. Prove that if f is Lipschitz on a particular domain, then f is uniformly continuous on that domain.

6. Prove that if the derivative of a function f exists and is bounded on some domain, that is $|f'(x)|\leq M$ for some M, then f is Lipschitz on that domain. How does the Lipschitz constant relate to the bound M?

7. Does the reverse hold true? That is, if f is Lipschitz, must the derivative (provided it exists) be bounded? Prove or give a counterexample.

8. Let $(f_n)$ be a pointwise convergent sequence of Lipschitz functions on $[0,1]$ with Lipschitz constant C. Prove that the sequence converges uniformly. Will the limit $\lim_{n\to\infty} f_n$ also be Lipschitz?

9. Let $x,y$ be fixed. Suppose f is Lipschitz with constant C. Define

(3)

\begin{align} f_1(x)=f(x), \quad f_2(x)=f(f(x)), \quad \ldots, \quad f_n(x)=f(f(\cdots f(x)\cdots)). \end{align}

What can you say about the limits of $f_n(x)$ and $f_n(y)$ for $n\to\infty$ if (a) $C<1$ or (b) $C\geq 1$?